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In-Depth Information
Ta b l e 2 . 5
Combinatorial Schema of Polymerase Chain Reaction
γζδ
γ
→
γζδ
λ
,
λ
template denaturation
c
ζ
c
δ
c
γ
c
ζ
c
δ
c
γζδ
λ
,
γ
λ
,
γ
γ
γζδ
δ
λ
λ
δ
,
→
,
primer hybridization
γ
c
ζ
c
δ
c
c
c
ζ
c
δ
c
c
γ
γ
γζδ
δ
γζδ
γ
γζδ
γ
,
→
,
polymerase extension
c
ζ
c
δ
c
c
c
ζ
c
δ
c
c
ζ
c
δ
c
mentioning when its value is not essential in the discussion). It is well-known that if
Type
¯
(
P
)=
{<
γϕδ
>,
γ
,
δ
}
,then
PCR
(
P
,
n
)
generates an
exponential amplification
of the
target molecule
, that is, copies of this molecule in a number that
is exponential with respect to the number
n
of steps, and only a minor quantity of
strands in
P
have types different from
<
γϕδ
>
.
Polymerase extension
ext
, depicted in Fig. 2.21, of a single string
<
γϕδ
>
αγ
, according
to a template
η
,isdefinedas:
¯
ext
(
αγ
,
β
γ
)=
αγβ
¯
(assuming that
γ
does not occur as a substring of
α
,and
β
), that is:
λ
ext
(
αγ
,
c
)=
αγβ
γ
c
β
If
.
It is useful to consider another extension operation, which we continue to denote
by
ext
, having a double string as argument, by setting (if the argument has a form
different from that here considered, this
ext
leaves it unchanged):
αγ
][
η
,thenweset
ext
(
αγ
,
η
)=
αγ
αγ
γ
c
)=
αγβ
ext
(
c
.
c
c
c
β
α
γ
β
PCR processes are easily representable by diagrams like that in Fig. 2.34, where
dotted arrows represent
ext
operation performed by polymerase enzymes.
All the possible PCR diagrams which result from the different forms of target
strings and from the different positions where primers may hybridize, are close to
100. Therefore, a natural question arises: in which cases does PCR provide exponen-
tial amplification? And, in these cases, is it possible to characterize, in general terms,
the form of strings that are exponentially amplified? Our DNA notation allows us to
answer these questions with the following lemma, which identifies important cases
of exponential amplification by PCR processes.
We say that single DNA strings
overlap
when they hybridize according
to the pattern depicted in Fig. 2.35. We define their
overlap concatenation
,by
setting:
ϕ
and
ψ
(
ϕ
ψ
<
ϕ
> <
ψ
>
=
ext
c
)
